Directly computing linear mass transport coefficients in stochastic models entails integrating over time the equilibrium correlations between atomic displacements. Here, we show how to improve the accuracy of kinetic Monte Carlo simulations via correlation splitting and conditioning, which statistically amounts to estimating the mass transport coefficients through a law of total diffusion. We illustrate the approach with kinetic path sampling simulations of atomic diffusion in a random alloy model in which percolating solute clusters trap the mediating vacancy. There, Green functions serve to generate first-passage paths escaping the traps and to propagate the long-time dynamics. When they also serve to estimate mean-squared displacements via conditioning, colossal reductions of statistical errors are achieved.